Assessing a hydrodynamic description for instabilities in highly dissipative, freely cooling granular gases

Abstract

An intriguing phenomenon displayed by granular flows and predicted by kinetic-theory-based models is the instability known as particle "clustering," which refers to the tendency of dissipative grains to form transient, loose regions of relatively high concentration. In this work, we assess a modified-Sonine approximation recently proposed [Garzó, Santos, and Montanero, Physica A 376, 94 (2007)] for a granular gas via an examination of system stability. In particular, we determine the critical length scale associated with the onset of two types of instabilities--vortices and clusters--via stability analyses of the Navier-Stokes-order hydrodynamic equations by using the expressions of the transport coefficients obtained from both the standard and the modified-Sonine approximations. We examine the impact of both Sonine approximations over a range of solids fraction φ<0.2 for small restitution coefficients e = 0.25-0.4, where the standard and modified theories exhibit discrepancies. The theoretical predictions for the critical length scales are compared to molecular dynamics (MD) simulations, of which a small percentage were not considered due to inelastic collapse. Results show excellent quantitative agreement between MD and the modified-Sonine theory, while the standard theory loses accuracy for this highly dissipative parameter space. The modified theory also remedies a high-dissipation qualitative mismatch between the standard theory and MD for the instability that forms more readily. Furthermore, the evolution of cluster size is briefly examined via MD, indicating that domain-size clusters may remain stable or halve in size, depending on system parameters.

Kinetic-theory predictions of clustering instabilities in granular flows: beyond the small-Knudsen-number regime

“…Despite such large Kn u at the onset of clustering, MD and continuum simulations are in excellent agreement for L cluster /d, suggesting the ability of hydrodynamics to describe the evolution of vortices and corresponding contribution to clustering. It is worth noting that previous works (Mitrano et al 2011(Mitrano et al , 2012 showing good agreement between MD simulations and KT-based predictions for the P. P. Mitrano and others onset of instabilities were limited to systems with relatively small gradients in all hydrodynamic variables.…”

“…Specifically, the particles simulated are frictionless yet inelastic and engage in binary, instantaneous collisions. Stability with respect to clustering is determined via analysis of the concentration Fourier spectra for varying domain sizes, as is further described in Mitrano et al (2012). Each domain size is tested for stability with ten replicate simulations that differ only by initial conditions.…”

Impact of collisional versus viscous dissipation on flow instabilities in gas–solid systems

“…This counterintuitive cross-over of kinetic energies is expected to provide a good quantitative test for kinetic-theory-based predictions of flow instabilities, which so far have been encouraging though limited to the onset of instabilities (Mitrano et al 2011(Mitrano et al , 2012. Comparisons with the evolution of flow instabilities is expected to be a more stringent test due to the large gradients (large Knudsen number) present in unstable flows.…”

“…The occurrence of instabilities 727 R2-2 Impact of collisional versus viscous dissipation on flow instabilities during the simulations was detected via a Fourier analysis of the particle concentration and velocity fields. This analysis was first accomplished for two-dimensional systems by Goldhirsch, Tan & Zanetti (1993), and was later extended to three-dimensional systems (Mitrano et al 2012). The same three-dimensional method was used to detect the instabilities in both the granular and gas-solid simulations reported here.…”

The surprising relevance of a continuum description to granular clusters

“…To show this, they juxtapose results from simulations and a numerical integration of the hydrodynamic equations that Garzó & Dufty (1999) derived for small Kn (figure 1). The linear stability analysis of Garzó (2005) yields a scale that Mitrano et al (2012) had interpreted as a cluster length. As Brilliantov et al (2004) suggested, clusters materialize in systems large enough to contain them.…”